Diffusion of Lambda(c) in hot hadronic medium and its impact on Lambda(c)/D ratio

Diffusion of

Λ

in hot hadronic medium and its impact on

Λ

=

D

ratio

Sabyasachi Ghosh,1 Santosh K. Das,2,3 Vincenzo Greco,2,3Sourav Sarkar,4 and Jan-e Alam4

1_{Instituto de Fsica Terica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271 Bloco II,} 01140-070 Sao Paulo, SP, Brazil

2_{Department of Physics and Astronomy, University of Catania, Via S. Sofia 64, 1-95125 Catania, Italy} 3

Laboratori Nazionali del Sud, INFN-LNS, Via S. Sofia 62, I-95123 Catania, Italy 4_{Theoretical Physics Division, Variable Energy Cyclotron Centre,}

1/AF, Bidhan Nagar, Kolkata 700064, India

(Received 24 July 2014; published 17 September 2014)

The drag and diffusion coefficients of theΛcð2286MeVÞhave been evaluated in the hadronic medium which is expected to be formed in the later stages of the evolving fireball produced in heavy ion collisions at RHIC and LHC energies. The interactions between the Λc and the hadrons in the medium have been derived from an effective hadronic Lagrangian, as well as from the scattering lengths obtained in the framework of heavy baryon chiral perturbation theory (HBχPT). In both the approaches, the magnitude of the transport coefficients turn out to be significant. A larger value is obtained in the former approach with respect to the latter. Significant values of the coefficients indicate a substantial amount of interaction of the Λcwith the hadronic thermal bath. Furthermore, the transport coefficients of theΛcare found to be different from the transport coefficients of theDmeson. The present study indicates that the hadronic medium has a significant impact on theΛc=D ratio in heavy ion collisions.

DOI:10.1103/PhysRevD.90.054018 PACS numbers: 12.38.Mh, 24.85.+p, 25.75.-q, 25.75.Nq

I. INTRODUCTION

One of the primary aims of the ongoing nuclear collision programs at the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies is to create a new state of matter known as quark gluon plasma (QGP), the bulk properties of which are governed by the light quarks and gluons. Heavy quarks (HQs≡charm and beauty) play crucial roles in understanding the properties of QGP [1], because they can witness the entire space-time evolution of the system as they are produced in the initial hard collision and remain extant during the evolution. Heavy flavor as a probe of the medium has generated significant interest in the recent past due to the suppression of its momentum distribution at large momentum in the thermal medium, denoted by RAAðpTÞ [2–5] and its elliptic flow (v2) [4]. Several attempts have been made to study these factors within the framework of the Fokker-Plank equation [1,6–18] and Boltzmann equation [19–22]. However, the

roles of hadronic phase have been ignored in these works. In heavy ion collision (HIC) at ultrarelativistic energies, the appearance of the hadronic phase is inevitable. To make reliable characterization of the QGP, the role of the hadronic phase should be assessed and its contribution must be subtracted out from the data. Recently the diffusion coefficients of theDandBmesons have been evaluated in the hadronic phase [23–30] and their effects on RAAðpTÞ

at large transverse momentum (pT)[31] and elliptic flow

(v2)[32,33]have been studied and found to be significant. Apart from the heavy mesons (DandB) the heavy baryon (_Λc) is also significant as its enhancement [34,35] due

to quark coalescence would affect the RAAðpTÞ of

nonphotonic electrons. Furthermore, the baryon-to-meson ratio, (_Λc=D), is fundamental for the understanding of

in-medium hadronization[36]with respect to the light quark sector [37]. Enhancement of the heavy baryon-to-meson ratio (_Λc=D) in AuþAu collisions compared to pþp

collisions affects the nonphotonic electron spectrum (RAA)

[38–41]. The branching ratio for the process Λc →eþ X_ð4_.5%1_.7%_{Þ is smaller than D→eþXð}17_.2% 1_.9%_{Þ, resulting in fewer electrons from decays ofΛcthan}

D. Hence, enhancement of the _Λc=D ratio in AuþAu

collision will reduce the observed nonphotonic electrons. We notice that thepT dependence of theΛc=Dratio may

get modified further in the hadronic medium as the interactions of_Λc andDwith hadrons are non-negligible.

Keeping this in mind, we attempt to study the transport coefficients (drag and diffusion coefficients) of _Λc in

hadronic phase.

The paper is organized as follows. In the next section we discuss the formalism used to evaluate the drag and diffusion coefficients of the heavy flavored baryon in a hot hadronic matter. Section III is devoted to presenting the results. SectionIVcontains summary and discussions.

II. FORMALISM

The drag and diffusion coefficients of the charmed baryon _Λc, propagating through a hot hadronic medium

kaons, andηmesons. The temperature of the bath can vary fromTcð∼170ÞtoTFð∼120MeVÞ, relevant for heavy ion

collisions at RHIC and LHC energies. Here Tc is the

transition temperature at which the QGP formed in HIC makes a transition to hadrons and TF is the freeze-out

temperature at which the hadrons cease to interact. In this temperature range the abundance ofΛcandDis small and

their thermal production and annihilation can be ignored. Hence, in evaluating the drag and diffusion coefficients of the _Λc only elastic processes will be considered.

For the elastic scattering of_Λc of momentump1with a thermal hadron, H of momentump2, i.e., for the process

Λcðp1Þ þHðp2Þ→Λcðp3Þ þHðp4Þ, the drag coefficient γ can be expressed as[43] (see also[44,45])

γ_¼piAi=p2 ð1Þ

where Aitakes the form

Ai¼ 1

2_Ep

1

Z _d3

p2 ð2_πÞ3_E

p2

Z _d3

p3 ð2_πÞ3_E

p3 Z _d3

p4 ð2_πÞ3_E

p4

× 1 g_Λc

X

jM_j2 ð2_πÞ4

δ4

ðp1þp2−p3−p4Þ

×f_ðp2Þf1fðp4Þg≡⟪ðp1−p3Þ⟫; ð2Þ

where g_Λc denotes the statistical degeneracy of _Λc, and f_ðp2Þ is the Bose-Einstein (BE) or Fermi-Dirac (FD) distribution function depending upon the spin of H in the initial channel. Similarly, the factor 1_fðp4Þ

repre-sents Bose enhanced or Pauli suppressed probability of the correspondingH in the final channel._jM_j2

represents the modulus square of the spin averaged matrix element for the ΛcþH elastic scattering process. Equation (2)illustrates

that the drag coefficient is the measure of the thermal average of the momentum transfer, p1−p3, weighted by the interaction through _jM_j2

.

Similarly the diffusion coefficientBcan be expressed as

B_¼1

4

⟪p2

3⟫−

⟪ðp1·p3Þ 2

⟫ p2

1

: _ð3_Þ

With an appropriate choice ofT_ðp3Þ, both the drag and diffusion coefficients can be expressed in a single equation as follows:

≪T_ðp1Þ≫¼

1

512_π4

1

Ep1 Z ∞

0

p2

2dp2dðcosχÞ

Ep2

×fˆðp2Þf1fðp4Þg λ12ðs;m2

p1;m 2

p2Þ ffiffiffi s

p

× Z −1

1

d_ðcosθc:m:Þ 1

g X

jM_j2

Z 2_π

0

dϕc:m:Tðp3Þ;

ð4_Þ

where λ_ðs;m2

p1;m 2

p2Þ¼fs−ðmp1þmp2Þ 2

gfs−ðm_p

1−mp2Þ 2

g

is the triangular function.

The drag and diffusion coefficients of _Λc can be

evaluated in hadronic matter by using_jTj2

[27] in place of _jM_j2

in Eq. (4), where the momentum independent T-matrix elements simply estimate the strength of the_Λc

interactions with the thermal hadrons.

The scattering lengths of _Λc with light pseudoscalar mesonsM_¼π,K,K¯, andηhave recently been obtained by Liu et al. [42] in the framework of HBχPT. From the scattering lengths,a(say) of_Λcinteracting withM, we can

extract the T-matrix element by using the relation

T_¼4_πðm

ΛcþmMÞa; ð5Þ

wherem_Λc andmMare the masses ofΛcand mesons (M),

respectively. From the scattering lengths (a in fm), the extracted values ofT are given in TableI.

In Ref.[42]Liuet al.have fixed the low energy constants (LECs) with the help of relations based on quark model symmetry, heavy quark spin symmetry, SUð4_{Þ flavor}

symmetry, and some empirical relations. However, a dimensionless constant α0 remains unknown, which is taken in the natural range _½−1_;1_{Š [42,46]. Therefore, the}

table shows a band of numerical values of T matrices, which are corrected up to the third order [O_ðϵ3

Þ] with the explicit power counting in HBχPT. Using theT matrices from Table I and corresponding BE distributions for H_¼M_¼π, K, η in Eq. (4), we can get an estimate of drag and diffusion coefficients of_Λc in hadronic matter.

Besides the scattering length approach, we have also investigated the contributions of the drag and diffusion coefficients, resulting from the Born-like scat-tering: _Λcπ→Σc →Λcπ. Using the effective hadronic

Lagrangian [47],

TABLE I. Table showing the extracted values of theTmatrix from the scattering length,a, which were obtained by Liuet al.[42].

ΛcM a(fm) T

Λcπ 0.06 9.28

ΛcK −0.0320.038 −12.42 to 1.06

L

ΛcΣcπ¼ g mπ

Λcγ5γμTrð~τ·Σ~c~τ·π~Þ þH:c:; ð6Þ

we can calculate the matrix elements for Λcπ scattering

diagrams viaΣc. The Lagrangian is based on the gauged

SU_ð4_{Þ flavor symmetry but with empirical masses. The}

coupling constant g¼0_.37 _{is taken from Ref. [47],}

where SU_ð4_{Þ relations are used to fix it.}

The possible s and uchannel diagrams for Λcþπ→

Σc→Λcþπprocesses are shown in the panels (a) and (b)

of Fig. 1. The matrix elements for the two channels are, respectively, given by

MΛcs π¼−

_2g

mπ 2

¯

u_ðp3Þγ5p4

ðp1þp2þm_ΣcÞ ðs−m2

ΣcÞ

γ5_p 2uðp1Þ

ð7_Þ

and

MΛcu π¼−

_2g

mπ 2

¯

u_ðp3Þγ 5

p2

ðp1−p4þm_ΣcÞ ðu−m2_Σ

cÞ

γ5p4uðp1Þ

:

ð8_Þ

Similarly from the Lagrangian density[47],

L_ΛcND¼ f

mD ¯

Nγ5 γμ_Λ

c∂μDþ∂μD¯ Λcγ5γμN; ð9Þ

one can obtain the matrix elements for the processes ΛcN→D→ΛcN and ΛcN¯ →D→ΛcN¯ (see Fig. 1).

The modulus square of the spin averaged total amplitudes

jM_j2 for all processes are given in the Appendix. Using those_jM_j2

from the effective hadronic model as well as the corresponding BE and FD distributions forH_¼πandNin Eq. (4), we can get alternative estimates of the drag and diffusion coefficients of Λc in the hadronic medium.

We have included form factors in each of the interaction vertices to take into account the finite size of the hadrons. For the u and s channel diagrams the form factors are taken as [47] Fu¼Λ2=ðΛ2þ~q2Þ and Fs¼Λ2= ðΛ2

þ!pi2Þ, respectively, where ~q is the three-momentum

transfer, pi is the initial momentum of the pions,

and _Λ¼1 _GeV.

III. RESULTS AND DISCUSSION

Let us first discuss the results of the drag coefficients obtained from the T-matrix elements of _ΛcM scattering,

given in TableI. The variation of the drag coefficient of_Λc

with temperature is depicted in Fig.2and compared with the drag coefficient of theDmesons[27]while propagating through the same thermal medium consisting of pions, kaons, and eta. The magnitude of the drag coefficients is quite significant, indicating a substantial interaction of_Λc with the thermal hadrons. The maximum and minimum values of the drag coefficient forΛccorrespond to the band

associated with theT-matrix element presented in TableI. The average value of the drag of_Λc is found to be smaller

than that ofD.

The single electron spectrum originating from the decays ofΛcandDmeasured in HIC is sensitive to the following

two mechanisms: (i) the production of Λc in HIC is

enhanced compared to that in pp because of the direct interaction of c with _½udŠ bound states available in the QGP [34], (ii) the Λc has a smaller branching ratio

to semileptonic decay than D. These two mechanisms lead to a deficiency of electrons at intermediate pT

[2_{< pT ðGeVÞ<}5_{][38]. If the drag ofΛc is more (less)}

thanD, then that will further reduce (enhance) the electrons in this domain ofpT. We find here that the value of the drag

of _Λc has a band of uncertainties as shown in Fig. 2;

therefore, it is not possible to draw a conclusion regarding which way the drag of _Λc will contribute to the electron spectra originating from the decays of charm mesons and baryons. However, measurements ofDmeson spectra via hadronic and semileptonic channels in the same collision conditions will help in estimating the electron spectra from Λc and hence its drag coefficients.

For a hadronic system of lifetime, Δτ, and drag γ, the momentum suppression is approximately given byRAA∼

e−Δτγ _{[24]. Picking up a value ofγ ofDatT ¼170MeV}

(A) (B)

(p1)

(p2)

(p3)

(p4)

(p1)

(p2) (p3)

(p4)

_

_

FIG. 1. Feynman diagrams for the scattering of Λc with the pion, nucleon, and antinucleon in the medium.

0.1 0.12 0.14 0.16

T(GeV)

0 0.01 0.02 0.03

γ

(fm

-1 )

Lambda_c-M (Max.) D Meson-M Lambda_c-M (Avg.) Lambda_c-M (Min.)

FIG. 2. Variation of the drag coefficient with temperature for the Λc and D meson [27] in a mesonic medium, where their interaction strengths are governed from their scattering lengths (SL)[42].

from the results displayed in Fig.2, we getRAA∼14%for Δτ_¼5_{fm=c. The values of RAA for Λc at the same}

temperature and Δτ are about 16% and 4%, respectively, for maximum and minimum values ofγ shown in Fig. 2. Similarly theΛc=Dratio at the same temperature andΔτis

approximately given byΛc=D∼eΔτðγD−γΛcÞ, whereγ_D and

γ_Λc are the drag coefficients of the D meson and _Λc,

respectively. The_Λc=Dratio can vary up to 12% depending on the minimum to maximum value of the drag coefficients of Λc.

The temperature variation of the drag coefficient of Λc

in a pionic medium has been depicted in Fig. 3. Here EL corresponds to the matrix element obtained from the effective hadronic Lagrangian[47]and SL corresponds to the scattering length or theT-matrix element obtained from the HBχPT. We found that the drag ofΛcin pionic medium

for EL is much larger than that for SL.

The corresponding SL results for the momentum dif-fusion coefficient as a function of temperature are depicted in Fig. 4. The difference between the maximum and minimum values of both coefficients gets larger at higher temperature as displayed in Figs.2 and4.

The variation of the ratio of_ΛctoDis shown in Fig.5as

a function of pT. The Fokker-Planck (FP) equation has

been used to study the time evolution of theDand_Λcin the hadronic bath of the equilibrated degrees of freedom. This is given by [31,43]

∂_f ∂_t ¼

∂ ∂_pi

AiðpÞfþ

∂

∂_pj½BijðpÞfŠ

; _ð10_Þ

where f is the momentum distribution of the nonequili-brated degrees of freedom, andAiðpÞandBijðpÞare related

to the drag and diffusion coefficients. The interaction between the probe and the thermal bath enter through the drag and diffusion coefficients. The initial distributions

of theD meson and _Λc are obtained (at the end of QGP phase) by using the fragmentation and coalescence tech-niques of[36]and results of[8]. Their ratio (at the end of the QGP phase) has been shown in the solid line of Fig.5. In the present calculation ofΛcspectra, resonances are not

taken into account at variance with Ref.[35].

The ratio estimated after evolving theD[31]andΛc in

the hadronic medium through the Fokker-Planck equation is displayed in Fig.5. In Fig.5QGP refers to the ratio at the end of the QGP phase and “_{QGPþhadronic}” _{refers to} the ratio at the end of the hadronic phase. Maximum and minimum of the ratio correspond to the maximum and minimum values of the drag and diffusion coefficients of Λc. The results indicate that the ratio gets enhanced for 2≤_pT ≤7_{due to the interactions of theDandΛc while}

propagating through the hadronic medium. Such enhance-ment will have interesting consequences on the nuclear suppression of the charm quarks in QGP measured through the single electron spectra originating from the decays of charmed hadrons.

0.1 0.12 0.14 0.16

T(GeV)

0 0.005

0.01 0.015 0.02

γ

(fm

-1 )

Lambda_c-Pi 10*Lambda_c-N 10*Lambda_c-Pion (SL)

FIG. 3. Variation of the drag coefficient with temperature forΛc in a pionic medium, using the dynamics of effective Lagrangian (EL) and scattering length (SL). The contribution of the drag coefficient for ΛcN scattering in the EL dynamics is also presented.

0.1 0.12 0.14 0.16

T(GeV)

0 0.005 0.01

B (GeV

2 /fm)

Lambda_c-M (Max) Lambda_c-M (Min.)

FIG. 4. Variation of the diffusion coefficient with temperature, whenΛcinteracts with all the light pseudoscalar mesonsM(in SL dynamics).

2 4 6 8

p_T(GeV) 0

0.1 0.2 0.3 0.4 0.5

Λ

c

/D

QGP+Hadronic (Minimum) QGP

QGP+Hadronic (Maximum)

IV. SUMMARY AND DISCUSSION

We have studied the diffusion of Λc in a hot hadronic

medium. Using scattering amplitudes, obtained by Liu et al.[42]in the framework of HBχPT, we have evaluated the drag and diffusion coefficients of theΛcinteracting with

a hadronic background composed of pions, kaons, and eta. We have also calculated the drag coefficients of the Λc

interacting with the pion and nucleon, using an effective hadronic Lagrangian. It is found that the coefficients in the pionic medium, obtained from the effective hadronic Lagrangian, are quite higher than those obtained from the dynamics of scattering length. However, the coefficients resulting from the ΛcN scattering obtained within an

effective hadronic model approach are comparable to the coefficients estimated in the scattering length approach. The value obtained for theΛc has been compared with the drag

coefficient of theDmeson calculated within the framework of heavy meson chiral perturbation theory. It is found that the value of the drag coefficient ofΛc is generally lower than

that ofDmesons. This result shows a significant effect on thepT dependence of theΛc=Dratio and hence also onRAA

of single electrons originating from the decay ofΛc.

ACKNOWLEDGMENTS

S. G. is thankful to Z. W. Liu and S. L. Zhu for their useful suggestions via email. S. G. is supported by FAPESP, Grant No. 2012/16766-0. S. K. D. and V. G. acknowledge the support by the ERC StG under the QGPDyn Grant No. 259684.

APPENDIX

The modulus square of the spin averaged total amplitude for the processes of Λcþπ→Σc→Λcþπ is given by

jMΛcπ_j2 ¼3₂

₂ g m_π

4 Ass ðs−m2_Σ

cÞ

2þ

Auu ðu−m2_Σ

cÞ

2

þ 2Asu

ðs−m2_Σ

cÞðu−m

2

ΣcÞ

; ðA1_Þ

where

Ass¼ ½−2m2πmΛcðs−m2ΛcÞ

2

ðm_Λcþ2_mΣ

cÞ

þm4

πðsþm

2

Λcþ2mΛcmΣcÞ

2

−ðs−m2

ΛcÞ

2

ðsu−m4

Λcþtm

2

Σcފ; ðA2Þ Auu¼ ½−2m2πmΛcðu−m2ΛcÞ

2

ðm_Λcþ2_mΣ

cÞ

þm4

πðuþm2_Λcþ2mΛcmΣcÞ

2

−_ðu−m2

ΛcÞ

2

ðsu−m4

Λcþtm

2

Σcފ; ðA3Þ

and

Asu¼ ½−4m6πm

2

Λcþ ð4s−2tþ4uÞm

6

Λc−2m

8

Λc

þ2_tðs2_−t2_−2su þu2

Þm_Λcm_Σcþ8_t2_m3

ΛcmΣc

þm4

Λcf−3s

2 þ5_t2

þ2_sðt−3_{uÞ þ}2_tu−3_u2

−2_tm2

Σcg

−ðs2−t2þu2Þðsu−tm2_Σ

cÞ þm

2

Λcfs

3 þ6_t2

m2

Σc

−_ðt−u_Þðt_þu_Þ2_þs2_ðt_þ3_uÞ−s_ðt2_þ4_tu−3_u2_Þg

þ2_m4

πfsuþ3m

4

Λc−4tmΛcmΣcþ8m

3

ΛcmΣc

−2_tm2

Σc−m

2

Λcðtþ4m

2

ΣcÞg−2m

2

πfmΛcmΣcfs

2

−4_t2

þs_ðt−2_{uÞ þtuþu}2

g þ14_tm3

ΛcmΣc

þ8_m4

Λcðtþm

2

ΣcÞ þ2tðsu−tm

2

ΣcÞ þm

2

Λcfs

2_−2t2

−tu_þu2−s_ðt_þ2_{uÞ þ ð}−4_sþ6_t−4_uÞm2

ΣcggŠ:

ðA4_Þ

The modulus square of the spin averaged total amplitude for the processes of _ΛcþN→D→ΛcþN and Λcþ

¯

N→D→ΛcþN¯ are, respectively, given by

jMΛcN j2

¼1₂

_f

mD

4 ₁

ðs−m2_DÞ2

× Tr_½ðp3þm_ΛcÞðp1þp2Þðp4−mNÞðp1þp2ފ

× Tr_½ðp2−mNÞðp1þp2Þðp1þm_ΛcÞðp1þp2ފ

¼2ðf=mDÞ 4

ðs−m2_DÞ2ðmΛc−mNÞ 2

fs−ðm_Λcþm_NÞ2g

×_½3_ðm4

Λcþm

4

NÞ þ10m

2

Λcm

2

Nþ ðtþuÞ

2

−s2

−4_ðtþuÞðm2

Λcþm

2

NÞ þsðmΛc−mNÞ2Š

ðA5_Þ

and

jMΛc_N¯ j2

¼1₂

_f

mD

4 ₁

ðu−m2_DÞ2

× Tr_½ðp3þm_ΛcÞðp1−p4Þðp2þmNÞðp1−p4ފ

× Tr_½ðp1þm_ΛcÞðp1−p4Þðp4þmNÞðp1−p4ފ

¼2ðf=mDÞ 4

ðu−m2_DÞ2ðmΛc−mNÞ 2

fu−ðm_Λcþm_NÞ2g

×_½3_ðm4

Λcþm

4

NÞ þ10m

2

Λcm

2

Nþ ðtþsÞ

2

−u2

−4_ðtþsÞðm2

Λcþm

2

NÞ þuðmΛc−mNÞ2Š:

ðA6_Þ

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